Sean Nixon

Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase transition point

Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase-transition point is analytically studied. A nonlinear Klein-Gordon equation is derived for the envelope of these wave packets. A variety of phenomena known to exist in this envelope equation are shown to also exist in the full equation, including wave blowup, periodic bound states, and solitary wave solutions.

Nonlinear wave dynamics near phase transition in PT-symmetric localized potentials

Abstract Nonlinear wave propagation in parity-time symmetric localized potentials is investigated analytically near a phase-transition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for a phase transition to occur are derived based on a generalization of the Krein signature. Using the multi-scale perturbation analysis, a reduced nonlinear ordinary differential equation (ODE) is derived for the amplitude of localized solutions near phase transition. Above the phase transition, this ODE predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodically-oscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below the phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.

Pyramid diffraction in parity-time-symmetric optical lattices

Nonlinear dynamics of wave packets in two-dimensional parity-time-symmetric optical lattices near the phase transition point are analytically studied. A fourth-order equation is derived for the envelope of these wave packets. A pyramid diffraction pattern is demonstrated in both the linear and nonlinear regimes. Blow-up is also possible in the nonlinear regime for both focusing and defocusing nonlinearities.

Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials

Abstract Stability of soliton families in one-dimensional nonlinear Schrodinger equations with non-parity-time ( PT )-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition. In addition, a pseudo-Hamiltonian–Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian–Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian. The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets ( λ , − λ , λ ⁎ , − λ ⁎ ) , similar to conservative systems and PT -symmetric systems. This quartet eigenvalue symmetry is very surprising for non- PT -symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.

Light propagation in periodically modulated complex waveguides

Light propagation in optical waveguides with periodically modulated index of refraction and alternating gain and loss are investigated for linear and nonlinear systems. Based on a multiscale perturbation analysis, it is shown that for many non-parity-time (

Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides

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Stable utility design for distributed resource allocation

) symmetric waveguides, their linear spectrum is partially complex, thus light exponentially grows or decays upon propagation, and this growth or delay is not altered by nonlinearity. However, several classes of non-

Exponential Asymptotics for Line Solitons in Two-Dimensional Periodic Potentials

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Destabilization of solitons in PT-symmetric optical lattices

-symmetric waveguides are also identified to possess all-real linear spectrum. In the nonlinear regime longitudinally periodic and transversely quasi-localized modes are found for

Bifurcation of Soliton Families from Linear Modes in Non‐PT‐Symmetric Complex Potentials

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